Unit 6 Quadrilaterals

Properties of Quadrilaterals Lesson 6.1 Properties of Quadrilaterals

Lesson 6.1 Objectives Identify a figure to be a quadrilateral. Use the sum of the interior angles of a quadrilateral. (G1.4.1)

Definition of a Quadrilateral A quadrilateral is any four-sided figure with the following properties: All sides must be line segments. Each side must intersect only two other sides. One at each of its endpoints, so that there are no: Gaps that do not connect one side to another, or Tails that extend beyond another side.

Example 6.1 Determine if the figure is a quadrilateral. Yes No No Yes Too many intersecting segments No Yes No gaps Yes No Too many sides No tails No No No curves

Interior Angles Recall that the interior angles of any figure are located in the interior and are formed by the sides of the figure itself. Review: How many degrees does a straight line measure? Review: What do you think the sum of the interior angles of a quadrilateral might be? Review: What is the sum of the interior angles of any triangle?

Theorem 6.1: Interior Angles of a Quadrilateral Theorem The sum of the measures of the interior angles of a quadrilateral is 360o. 1 2 3 4 m 1 +m 2 + m 3 + m 4 = 360o

Example 6.2 Find the missing angle.

Example 6.3 Find the x.

Lesson 6.1 Homework Lesson 6.1 – Properties of Quadrilaterals Due Tomorrow

Lesson 6.2 Day 1: Parallelograms

Lesson 6.2 Objectives Define a parallelogram Define special parallelograms Identify properties of parallelograms (G1.4.3) Use properties of parallelograms to determine unknown quantities of the parallelogram (G1.4.4)

Definition of a Parallelogram A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

Theorem 6.2: Congruent Sides of a Parallelogram If a quadrilateral is a parallelogram, then its opposite sides are congruent. The converse is also true! Theorem 6.6

Theorem 6.3: Opposite Angles of a Parallelogram If a quadrilateral is a parallelogram, then its opposite angles are congruent. The converse is also true! Theorem 6.7

Example 6.4 x = 11 c – 5 = 20 m = 101 c = 25 y = 8 d + 15 = 68 d = 53 Find the missing variables in the parallelograms.      x = 11 c – 5 = 20 m = 101 c = 25 y = 8 d + 15 = 68 d = 53

Theorem 6.4: Consecutive Angles of a Parallelogram If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. The converse is also true! Theorem 6.8 Q P R S m P + m S = 180o m Q + m R = 180o m P + m Q = 180o m R + m S = 180o

Theorem 6.5: Diagonals of a Parallelogram If a quadrilateral is a parallelogram, then its diagonals bisect each other. Remember that means to cut into two congruent segments. And again, the converse is also true! Theorem 6.9

Example 6.5 Find the indicated measure in  HIJK HI GH KH HJ m KIH 16 Theorem 6.2 GH 8 Theorem 6.5 KH 10 HJ Theorem 6.5 & Seg Add Post m KIH 28o AIA Theorem m JIH 96o Theorem 6.4 m KJI 84o Theorem 6.3

Theorem 6.10: Congruent Sides of a Parallelogram If a quadrilateral has one pair of opposite sides that are both congruent and parallel, then it is a parallelogram.

Example 6.6 Yes! Yes! Yes! Yes! Yes! Yes! Is there enough information to prove the quadrilaterals to be a parallelogram. If so, explain. Yes! Yes! Yes! Both pairs of opposite sides are congruent. (Theorem 6.6) One pair of parallel and congruent sides. (Theorem 6.10) Both pairs of opposite angles are congruent. (Theorem 6.7) Yes! Yes! Yes! OR One pair of parallel and congruent sides. (Theorem 6.10) Both pairs of opposite angles are congruent. (Theorem 6.7) All consecutive angles are supplementary. (Theorem 6.8) The diagonals bisect each other. (Theorem 6.9) Both pairs of opposite sides are congruent. (Theorem 6.6)

Lesson 6.2a Homework Lesson 6.2: Day 1 – Parallelograms Due Tomorrow

Day 2: (Special) Parallelograms Lesson 6.2 Day 2: (Special) Parallelograms

Rhombus A rhombus is a parallelogram with four congruent sides. The rhombus corollary states that a quadrilateral is a rhombus if and only if it has four congruent sides.

Theorem 6.11: Perpendicular Diagonals A parallelogram is a rhombus if and only if its diagonals are perpendicular.

Theorem 6.12: Opposite Angle Bisector A parallelogram is a rhombus iff each diagonal bisects a pair of opposite angles.

Rectangle A rectangle is a parallelogram with four congruent angles. The rectangle corollary states that a quadrilateral is a rectangle iff it has four right angles.

Theorem 6.13: Four Congruent Diagonals A parallelogram is a rectangle iff all four segments of the diagonals are congruent.

Square A square is a parallelogram with four congruent sides and four congruent angles.

Square Corollary A quadrilateral is a square iff its a rhombus and a rectangle. So that means that all the properties of rhombuses and rectangles work for a square at the same time.

Example 6.7 Classify the parallelogram. Explain your reasoning. Must be supplementary    Rhombus Rectangle Square Diagonals are perpendicular. Theorem 6.11 Diagonals are congruent. Theorem 6.13 Square Corollary

Lesson 6.2b Homework Lesson 6.2: Day 2 – Parallelograms Due Tomorrow

Lesson 6.3 Trapezoids and Kites

Lesson 6.3 Objectives Identify properties of a trapezoid. (G1.4.1) Recognize an isosceles trapezoid. (G1.4.1) Utilize the midsegment of a trapezoid to calculate other quantities from the trapezoid. Identify a kite. (G1.4.1)

Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called the bases. The nonparallel sides are called legs. The angles formed by the bases are called the base angles.

Example 6.8 Find the indicated angle measure of the trapezoid. CIA CIA Consecutive Interior Angles are supplementary! CIA CIA Recall that a trapezoid has one set of parallel bases.

Example 6.9 Find x in the trapezoid. CIA CIA Consecutive Interior Angles are supplementary! Find x in the trapezoid. CIA CIA

Isosceles Trapezoid If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.

Theorem 6.14: Bases Angles of a Trapezoid If a trapezoid is isosceles, then each pair of base angles is congruent. That means the top base angles are congruent. The bottom base angles are congruent. But they are not all congruent to each other!

Theorem 6.15: Base Angles of a Trapezoid Converse If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid.

Theorem 6.16: Congruent Diagonals of a Trapezoid A trapezoid is isosceles if and only if its diagonals are congruent. Notice this is the entire diagonal itself. Don’t worry about it being bisected cause it’s not!!

Example 6.10 Find the measures of the other three angles.  Supplementary because of CIA  127o 127o 83o   97o 83o Supplementary because of CIA

Midsegment The midsegment of a trapezoid is the segment that connects the midpoints of the legs of a trapezoid.

Theorem 6.17: Midsegment Theorem for Trapezoids The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. It is the average of the base lengths! A B C D M N

Or essentially double the midsegment! Example 6.11 Find the indicated length of the trapezoid. ? ? ? Multiply both sides by 2. Or essentially double the midsegment!

Kite A kite is a quadrilateral that has two pairs of consecutive sides that are congruent, but opposite sides are not congruent. It looks like the kite you got for your birthday when you were 5! There are no sides that are parallel.

Theorem 6.18: Diagonals of a Kite If a quadrilateral is a kite, then its diagonals are perpendicular.

Theorem 6.19: Opposite Angles of a Kite If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. The angles that are congruent are between the two different congruent sides. You could call those the shoulder angles. NOT

Example 6.12 Find the missing angle measures. But K  M 125o 64o But K  M 125o  88o 60 + K + 50 + M = 360 60 + M + 50 + M = 360 K = 88 110 + 2M = 360 88 + 120 + 88 + J = 360 2M = 250 M = 125 296 + J = 360 J = 64 K = 125

Example 6.13 Find the lengths of all the sides of the kite. Round your answer to the nearest hundredth. a2 + b2 = c2 52 + 52 = c2 a2 + b2 = c2 25 + 25 = c2 7.07 7.07 52 + 122 = c2 50 = c2 25 + 144 = c2 c = 7.07 169 = c2 13 13 c = 13 Use Pythagorean Theorem! Cause the diagonals are perpendicular!! a2 + b2 = c2

Lesson 6.3 Homework Lesson 6.3 – Trapezoids and Kites Due Tomorrow

Perimeter and Area of Quadrilaterals Lesson 6.4 Perimeter and Area of Quadrilaterals

Lesson 6.4 Objectives Find the perimeter of any type of quadrilateral. (G1.4.1) Find the area of any type of quadrilateral. (G1.4.3)

Postulate 22: Area of a Square Postulate The area of a square is the square of the length of its side. A = s2 s

Theorem 6.20: Area of a Rectangle The area of a rectangle is the product of a base and its corresponding height. Corresponding height indicates a segment perpendicular to the base to the opposite side. A = bh h b

Example 6.14 Find the perimeter and area of the given quadrilateral.

Theorem 6.21: Area of a Parallelogram The area of a parallelogram is the product of a base and its corresponding height. Remember the height must be perpendicular to one of the bases. The height will be given to you or you will need to find it. To find it, use Pythagorean Theorem a2 + b2 = c2 A = bh h b

Theorem 6.23: Area of a Trapezoid The area of a trapezoid is one half the product of the height and the sum of the bases. The height is the perpendicular segment between the bases of the trapezoid. A = ½ (b1+b2) h b1 h b2

Theorem 6.24: Area of a Kite d1 d2 The area of a kite is one half the product of the lengths of the diagonals. A = ½ d1d2 d1 d2

Theorem 6.25: Area of a Rhombus The area of a rhombus is equal to one half the product of the lengths of the diagonals. A = ½ d1d2 d1 d2

Example 6.15 Find the perimeter and area of the given quadrilateral.

Area Postulates Postulate 23: Area Congruence Postulate If two polygons are congruent, then they have the same area. Postulate 24: Area Addition Postulate The area of a region is the sum of the areas of its nonoverlapping parts.

Example 6.16 Find the perimeter and area of the given figure. Assume all corners form a right angle.

Lesson 6.4 Homework Lesson 6.4 – Perimeter and Area of Quadrilaterals Due Tomorrow

Special Quadrilaterals Lesson 6.5 Special Quadrilaterals

Lesson 6.6 Objectives Create a hierarchy of polygons Identify special quadrilaterals based on limited information

Polygon Hierarchy Polygons Pentagons Triangles Quadrilaterals Parallelogram Trapezoid Kite Rhombus Rectangle Isosceles Trapezoid Square

How to Read the Hierarchy NEVER How to Read the Hierarchy Polygons Triangles Quadrilaterals Pentagons Rhombus Rectangle Trapezoid Parallelogram Kite Square Isosceles Trapezoid ALWAYS SOMETIMES But a parallelogram is sometimes a rhombus and sometimes a square. So that means that a square is always a rhombus, a parallelogram, a quadrilateral, and a polygon. However, a parallelogram is never a trapezoid or a kite.

Using the Hierarchy Remember that a square must fit all the properties of its “ancestors.” That means the properties of a rhombus, rectangle, parallelogram, quadrilateral, and polygon must all be true! So when asked to identify a figure as specific as possible, test the properties working your way down the hierarchy. As soon as you find a figure that doesn’t work any more you should be able to identify the specific name of that figure.

Homework 6.6 HW p367-370 8-35, 55-65 Due Tomorrow Test Friday March 26